Solving large-scale security-constrained economic dispatch problem in real-time

ABSTRACT

A system, a method and a computer program product for determining an amount of an electric power to be generated in an electric power system and determining a total cost for generating the amount of electric power while satisfying at least one contingency constraint and one or more customer request. The system creates an optimization problem for calculating amount of the electric power to be generated and a total cost for generating the calculated amount of the electric power while meeting the at least one contingency constraint. The system runs the optimization problem in real-time. The system outputs, from the optimization problem, an output specifying the calculated amount of the electric power and the total cost to generate the calculated amount of the electric power.

RELATED APPLICATIONS

This application is a continuation application of U.S. application Ser.No. 13/183,865, filed Jul. 15, 2011.

BACKGROUND

The present application generally relates to dispatching energy, e.g.,an electric power. More particularly, the present application relates todetermining an amount of electric power to be generated and their costto generate while meeting at least one contingency constraint and one ormore customer requests.

Electricity markets in the United States are comprised of twointerconnected markets: a day ahead market and a real-time or balancingmarket. The day-ahead market focuses on an electricity power generationschedule per an electric power generator. The real-time market focuseson economic dispatch, i.e., determining an output of an electric powergenerator and its cost to generate the electric power.

Currently, linear direct current (DC) approximation of a (nonlinear,non-convex) AC (Alternating Current) power flow equation is mostly usedin the existing electric power generation systems. A reference to P. N.Biskas and A. G. Bakirtzis, entitled “A decentralized solution to thesecurity constrained DC-OPF problem of multi-area power systems,” PowerTech, 2005 IEEE Russia, pp. 1-7, June 2005, wholly incorporated byreference as if set forth herein, describes the linear direct currentapproximation of the AC power flow equation in detail. A main drawbackof the linear DC approximation is that it does not capture a physicalelectric power flow more realistically than the AC power flow equation.Currently, solving the AC power flow equation without the linear DCapproximation requires large amount of memory usage (e.g., more than oneTerabyte, etc.) and cannot be performed in real-time. Currently, thesolution of the AC optimal power flow problem is usually a low quality,i.e., far from an optimal solution (i.e., optimality referring to idealamount of electric power to be generated at an ideal cost to generatethe ideal amount of electric power). The linear DC approximationrequires trials and errors to ensure a feasibility of its solution anddoes not provide an optimal solution.

SUMMARY OF THE INVENTION

The present disclosure describes a system, method and computer programproduct for determining an amount of an electric power to generate in anelectric power system and determining a total cost for generating theamount of electric power while satisfying at least one contingencyconstraint and one or more customer request.

In one embodiment, there is provided a system for determining an amountof an electric power to generate in an electric power system anddetermining a total cost for generating the amount of electric powerwhile satisfying at least one contingency constraint and one or morecustomer request. The system includes, but is not limited to: at leastone memory device and at least one processor connected to the memorydevice. The system is configured to create an optimization problem forcalculating an amount of the electric power to be generated and a totalcost for generating the calculated amount of the electric power whilemeeting the at least one contingency constraint and the one or morecustomer request. The contingency constraint includes one or more of: atransmission line failure, a power generator failure, a transformerfailure, a structure of a power distribution network, and a topology ofthe power distribution network. The system is configured to run theoptimization problem in real-time. The system is configured to output,from the optimization problem, an output specifying the calculatedamount of the electric power and the total cost to generate thecalculated amount of the electric power.

In a further embodiment, to run the optimization problem, the system isconfigured to solve the optimization problem without considering the atleast one contingency constraint. The system is configured to checkwhether a result of the computed optimization problem satisfies the atleast one contingency constraint. The system is configured to output theresult in response to determining that the result satisfies the at leastone contingency constraint. The system is configured to prune, inresponse to determining that the result does not satisfy the at leastone contingency constraint, one or more results of the optimizationproblem that do not satisfy the at least one contingency constraint. Thesystem is configured to solve the optimization problem with the prunedresults.

In a further embodiment, to run the optimization problem, the system isconfigured to generate a template to solve the optimization problem. Thesystem is configured to reformulate the optimization problem based onthe generated template. The system is configured to form an augmentedLagrangian for the reformulated optimization problem. The system isconfigured to initialize Lagrangian multipliers and decision variablesaccording to the at least one contingency constraint. The system isconfigured to solve the reformulated optimization problem with theinitialized Lagrangian multipliers and decision variables, based on analternating direction method of multipliers.

In a further embodiment, the system is further configured to break thereformulated optimization problem into one or more sub-problems. Thesystem is configured to solve the one or more sub-problems with theinitialized Lagrangian multipliers and decision variables, based on theformed augmented Lagrangian. The system is configured to evaluatewhether a solution of the one or more sub-problems meets a stoppingcriterion. The system is configured to output the solution in responseto determining that the solution meets the stopping criterion. Thesystem is configured to update the Lagrangian multipliers based on thesolution, in response to determining that the solution does not meet thestopping criterion.

In a further embodiment, to run the optimization problem, the system isconfigured to build a tree data structure. Each node of the tree datastructure describes a possible solution of the optimization problem. Thesystem is configured to compute, at each node of the built tree datastructure, an upper bound and lower bound of an objective function. Theobjective function determines the cost for generating the electricpower. The lower bound is calculated from Lagrange duality. The upperbound is calculated from an interior-point method. The system isconfigured to evaluate, at each node of the built tree data structure,whether the computed upper bound and computed lower bound satisfy astopping criterion. The system is configured to output the computedupper bound and the computed lower bound in response to determining thatthe computed upper bound and computed lower bound satisfy the stoppingcriterion. The system is configured to apply at least one branchingprocedure on the tree data structure in response to determining that thecomputed upper bound and computed lower bound do not satisfy thestopping criteria. The system is configured to re-compute the lowerbound in a region in the tree data structure branched according to theapplied branching procedure. The system is configured to update theupper bound in the region in the tree data structure. The system isconfigured to delete nodes in the region based on the re-computed thelower bound and updated upper bound or an intersection of the branchedregion and an original feasible set. The original feasible set refers topossible solutions of the optimization problem satisfying the at leastone contingency constraint. The system is configured to repeat theevaluating, the outputting, the applying, the re-computing, theupdating, and the deleting.

In a further embodiment, the stopping criterion includes one or more of:a gap between the smallest computed lower bound and the computed upperbound is lower than a pre-determined threshold, or the tree datastructure is empty.

In a further embodiment, the one or more customer request includes: ademand or load for a certain amount of the electric power.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings are included to provide a furtherunderstanding of the present invention, and are incorporated in andconstitute a part of this specification.

FIG. 1 is a flow chart that describes method steps for solving anoptimization problem that calculates an amount of electric power to begenerated and a total cost for generating the calculated amount whilemeeting at least one contingency constraint and one or more customerrequest in one embodiment.

FIG. 2 is another flow chart that describes method steps for solving anoptimization problem that calculates an amount of the electric power tobe generated and a total cost for generating the calculated amount ofthe electric power while meeting at least one contingency constraint andthe one or more customer request in another embodiment.

FIG. 3 is another flow chart that describes method steps for solving anoptimization problem that calculates amount of electric power to begenerated and a total cost for generating the calculated amount electricpower while meeting at least one contingency constraint and the one ormore customer request in another embodiment.

FIG. 4 illustrates an exemplary hardware configuration for implementingthe flow charts depicted in FIGS. 1-3 in one embodiment.

FIG. 5 is a flow chart for determining an amount of an electric power togenerate in an electric power system and determining a total cost forgenerating the amount of electric power while satisfying at least onecontingency constraint and one or more customer request in oneembodiment.

DETAILED DESCRIPTION

In this disclosure, there are provided a method, a system and a computerprogram product for solving an AC (Alternative Current) electric optimalpower flow problem (e.g., problem (1) described below) in real-time togenerate an optimal solution. The optimal solution specifies an idealamount of electric power to be generated and a cost for generating theideal amount of the electric power while satisfying at least onecontingency constraint and one or more customer request. The contingencyconstraint includes one or more of: a transmission line failure, a powergenerator failure, a transformer failure, a structure of a powerdistribution network, and a topology of the power distribution network.The one or more customer request comprises: a demand or load for acertain amount of an electric power, etc. In one embodiment, thecustomer request, e.g., the demand, is incorporated in the definitionsof functions h, and g, in problem (1) described below.

In one embodiment, to solve the AC electric optimal power flow problem,a computing system (e.g., a computing system 400 shown in FIG. 4) isconfigured to apply method steps described in FIG. 1 or FIG. 2 or FIG. 3on the AC electric optimal power flow problem. Method steps described inFIGS. 1-2 are heuristics that generate sub-optimal solutions of the ACelectric optimal power flow problem. Method steps described in FIG. 3generate the optimal solution of the AC electric optimal power flowproblem. Benefits of these method steps described in FIGS. 1-3 include,but are not limited to: (1) these method steps can handle a nonlinearnon-convex optimization problem (e.g., AC electric optimal power flowproblem), which outputs a result that specifies one or more of: (a)amount of electric power to be generated to satisfy the contingencyconstraint and customers' requests; and (b) a lowest cost to generatethe amount of the electric power; (2) The computing system 400 or anytraditional computing system can run these method steps in FIGS. 1-3 inreal-time; (3) The computing system 400 or traditional computing systemcan run method steps in FIG. 1 and/or FIG. 2 and/or FIG. 3 in parallelto improve running time of these method steps; and (4) The output frommethod steps in FIG. 3 can be used to determine quality of outputs ofmethod steps in FIGS. 1-2 because the output from the method steps inFIG. 3 provides the optimal solution of the AC electric optimal powerflow problem.

In another embodiment, to solve the AC electric optimal power flowproblem, the computing system 400 is configured to utilize one or moreof: Benders decomposition algorithm, Alternating direction method ofmultipliers, and Branch-and-bound algorithm. A reference to A. M.Geoffrion, “Generalized Benders decomposition,” Journal of OptimizationTheory and Applications, vol. 10, no. 4, 1972, wholly incorporated byreference as set forth herein, describes Benders decomposition algorithmin detail. A reference to N. Parikh, et al., “Distributed optimizationand statistical learning via the alternating direction method ofmultipliers,” Jan. 27, 2011, wholly incorporated by reference as setforth herein, describes alternating direction method of multipliers indetail. A reference to Leo Liberti, “Introduction to globaloptimization,” Lix, Ecole Polytechnique, Feb. 15, 2008, whollyincorporated by reference as set forth herein, describesBranch-and-bound algorithm in detail.

By solving the AC electric optimal power flow problem, e.g., by runningthe method steps in FIG. 1 or FIG. 2 or FIG. 3, the computing system 400solves a security-constrained economic dispatch problem (i.e.,determining an amount of electric power to be generated under aconstraint) because the output of the method steps in FIG. 1 or FIG. 2or FIG. 3 specifies an amount of electric power to be generated per anelectric power generator and a cost to operate the generator and a costto generate the electric power under the contingency constraint. Byrunning method steps in FIG. 1 or FIG. 2 or FIG. 3, an electric powercompany (e.g., Con Edison®, etc.) can provide electric power reliably toall of its customers even though there exists an occurrence of atransmission line failure, an electric power generator failure or atransformer failure.

The AC electric optimal power flow problem can be formulated as follows:

$\begin{matrix}{{\min\;{f\left( u_{0} \right)}}{{s.t.{h_{c}\left( {x_{c},u_{c}} \right)}} = 0}{{g_{c}\left( {x_{c},u_{c}} \right)} \leq 0}{{{{u_{c}^{i} - u_{0}^{i}}} \leq {\Delta\; u_{i}^{\max}}},{i \in G},{c = 0},\ldots\mspace{14mu},C,}} & (1)\end{matrix}$where c=0 represents an electric power system at a normal case (i.e., nooccurrence of contingency constraint), c=1, . . . , C, is a variablerepresenting a contingency constraint, x_(c) is a vector of statevariables (e.g., complex voltages) for a c-th configuration, u_(c) is avector of control variables (e.g., the active and reactive powers,etc.), a generation cost f(u₀)=Σ_(iεG)(c_(0i)(u₀ ^(i))²+c_(1i)u₀^(i)+c_(2i)) with cost parameters c_(0i), c_(1i), c_(2i)≦0 (G is a setof generators), and C is the number of contingency constraints. Δu_(i)^(max) is a pre-determined maximal allowed variation of controlvariables, and h_(c) and g_(c) are operational constraints including ACpower flow balance equations (e.g., P₁+P₂+P₃=0, where P₁ represents anelectric power at an electric node “1,” P₂ represents an electric powerat an electric node “2,” P₃ represents an electric power at an electricnode “3.”). Note that the optimization problem (i.e., the problem (1))allows control variables to be reinitialized in order to satisfycontingency constraints. The problem (1) is advantageously solved inreal-time.

FIG. 5 is a flow chart that describes method steps for determining anamount of an electric power to generate in an electric power system anddetermining a total cost for generating the amount of electric powerwhile satisfying at least one contingency constraint and one or morecustomer request in one embodiment. At step 500, the computing system400 is configured to create the optimization problem (e.g., the problem(1) described above) for calculating an amount of the electric power tobe generated and a total cost for generating the calculated amount ofthe electric power while meeting the at least one contingency constraintand the one or more customer request. At step 510, the computing systemis configured to solve the optimization problem in real-time, e.g., byrunning the method steps in FIG. 1 or FIG. 2 or FIG. 3. The computingsystem is configured to output, from the optimization problem, an outputspecifying the calculated amount of the electric power and the totalcost to generate the calculated amount of the electric power.

By running method steps in FIG. 1, the computing system 400 isconfigured to decompose the optimization problem into a master problem(e.g., the problem (10) described below) and sub-problems (e.g.,problems (5) and (6) described below). The computing system 400 isconfigured to check a feasibility of a solution of each sub-problem forthe master problem. This feasibility check can be performed in parallel,e.g., by using two or more processors 411 in the computing system 400.Method steps in FIG. 1 can also solve the optimization problem even ifthe optimization problem is non-convex. By running method steps in FIG.2, the computing system 400 is configured to improve quality of thesolution from method steps in FIG. 1.

FIG. 1 is a flow chart that describes method steps for solving theoptimization problem in one embodiment. At step 110, the computingsystem 400 is configured to compute the optimization problem withoutconsidering the contingency constraint. In one embodiment, theoptimization problem (1) is expressed as a general nonlinear non-convexoptimization problem:

$\begin{matrix}{{\min\;{f(x)}}{{s.t.x} \in X}{{G(y)} \leq 0}{{H(y)} = 0}{{{Ax} + {By} + b} \leq 0}{{y^{L} \leq y \leq y^{U}},}} & (2)\end{matrix}$where X⊂P^(n) and yεP^(m), A and B are matrices. In one embodiment, f isa convex function, whereas G, H and X are non-convex, b is a constantvector, y^(L) is a lower bound of the variable y, and the y^(u) is anupper bound of the variable y.

An initial master problem is of the problem (2) is as follows:

$\begin{matrix}{{\min\;{f(x)}}{x \in {X.}}} & (3)\end{matrix}$

Let x be a solution of the initial master problem. Then, the computingsystem 400 is configured to check if the following subproblem isfeasibleG(y)≦0H(y)=0A x+By+b≦0y ^(L) ≦y≦y ^(U).  (4)Note that each contingency constraint associated with each c in theproblem (1) may be written in a form of problem (4).

Returning to FIG. 1, at step 120, the computing system is configured tocheck whether a result of the computed optimization problem (e.g.,problem (3)) satisfies the contingency constraint described above, e.g.,by running problem (4). Note that the optimization problem (1) can becompactly expressed as the problem (2). Each contingency constraintassociated with c in the problem (1) can be expressed in the form ofproblem (4). So, checking the contingency constraint for each c isequivalent to evaluating the feasibility of the problem (4). In responseto determining that the result satisfies the contingency constraint, thesystem is configured to output the result. In other words, if theequation (4) is feasible for the given x, at step 150, the computingsystem 400 terminates running the method steps in FIG. 1. Otherwise, atstep 130, the computing system 400 is configured to prune, in responseto determining that the result does not satisfy the contingencyconstraint, one or more results of the computed optimization problem(e.g., problem (3)) that do not satisfy the contingency constraint. Toprune the one or more results, the computing system 400 is configured togenerate a cut that is added to the initial master problem (e.g.,problem (3)).

To check the feasibility of the problem (4) or generate the cut, thecomputing system is configured to solve, for the fixed x, a sub-problem

$\begin{matrix}{{\min\limits_{y,\alpha_{i}}{\sum\limits_{i}\alpha_{i}}}{{subject}\mspace{14mu}{to}\text{:}}{{G(y)} \leq 0}{{H(y)} = 0}{{{A\overset{\_}{x}} + {By} + b - \alpha} \leq 0}{y^{L} \leq y \leq y^{U}}{{\alpha \geq 0},}} & (5)\end{matrix}$where α_(i) is a new slack variable.

Assume, in one embodiment, that an optimal result of the problem (5) isstrictly positive, i.e., the problem (4) is infeasible, i.e., for thegiven x, there is no y satisfying constraints (4). Further in oneembodiment, it is assumed that ( y, α) is an optimal solution of theproblem (5). Then, by substituting z≈y−y^(L) and using the Taylorexpansions of G(y) and H(y) at y−y^(L), the computing system isconfigured to obtain a relaxed linear programming:

$\begin{matrix}{{\min\limits_{\alpha,z}{\sum\limits_{i}\alpha_{i}}}{{{G\left( \overset{\_}{y} \right)} + {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {z - \left( {\overset{\_}{y} - y^{L}} \right)} \right)}} \leq 0}{{{H\left( \overset{\_}{y} \right)} + {{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {z - \left( {\overset{\_}{y} - y^{L}} \right)} \right)}} = 0}{{{Bz} - \alpha + {A\overset{\_}{x}} + b + {By}^{L}} \leq 0}{z \leq {y^{U} - y^{L}}}{\alpha,{z \geq 0.}}} & (6)\end{matrix}$

For a general optimization problem, the problem (5) can be stillinfeasible if the computing system is configured to introduce theauxiliary variable a for a constraint A x+By+b−α≦0. To construct a newoptimization problem whose feasible set is definitely non-empty, thecomputing system is configured to consider constraints G(y)≦0 andH(y)=0. However, in an electric power system analysis, for α is largeenough (e.g., larger than 1000, etc.), the problem (5) is feasible.

Assume that an optimal result of the problem (6) is strictly positive.The computing system 400 is configured to construct linear cuts (e.g.,equations (7) and (8)-(9) below) based on Lagrange multipliers arisingfrom the problems (5) or (6). Lagrange multipliers refer to a methodthat finds a maximum and a minimum of a function subject to constraints.Yan-Bin Jia, “Lagrange Multipliers,” Nov. 18, 2008, wholly incorporatedby reference as if set forth herein, describes Lagrange multipliers indetail. Note that if the optimal result of the problem (6) is strictlypositive, then (4) is infeasible, since the optimal result of theproblem (6) is less than or equal to the optimal result of the problem(5).

In solving the optimization problem under one or more constraints, it ispossible to convert the optimization problem (e.g., problem (6)) to adual problem (e.g., Lagrangian dual problem, etc.). The Lagrangian dualproblem is formulated by using Lagrangian multipliers to incorporate theconstraints to an objective function. A reference to Eric P. Xing andGuang Xiang, entitled “Optimal Margin Principle and Lagrangian Duality,”2007, Carnegie Mellon University, wholly incorporated by reference as ifset forth herein, describes Lagrangian dual problem in detail.

The Lagrange dual problem of the problem (6) is as follows:

${{{\max\limits_{{\pi \geq 0},{\mu \geq 0},\eta}{\min\limits_{\underset{0 \leq z \leq {y^{U} - y^{L}}}{0 \leq \alpha}}{\sum\limits_{i}\;\alpha_{i}}}} + {\pi^{T}\left( {{Bz} - \alpha + {A\overset{\_}{x}} + {By}^{L} + b} \right)} + {\eta^{T}\left( {{H\left( \overset{\_}{y} \right)} + {{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {z - \left( {\overset{\_}{y} - y^{L}} \right)} \right)}} \right)} + {\mu^{T}\left( {{G\left( \overset{\_}{y} \right)} + {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {z - \left( {\overset{\_}{y} - y^{L}} \right)} \right)}} \right)}} = {{{\max\limits_{{\pi \geq 0},{\mu \geq 0},\eta}{\min\limits_{\underset{0 \leq z \leq {y^{U} - y^{L}}}{0 \leq \alpha}}{\sum\limits_{i}\;{\alpha_{i}\left( {1 - \pi_{i}} \right)}}}} + {\left( {{B^{T}\pi} + {{\nabla_{y}^{T}{H\left( \overset{\_}{y} \right)}}\eta} + {{\nabla_{y}^{T}{G\left( \overset{\_}{y} \right)}}\mu}} \right)^{T}z} + {\pi^{T}\left( {{A\overset{\_}{x}} + {By}^{L} + b} \right)} - {\eta^{T}{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)} + {\mu^{T}\left( {{G\left( \overset{\_}{y} \right)} - {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)}} \right)}} = {{\max\limits_{{0 \leq \pi_{i} \leq 1},{\mu \geq 0}}{\pi^{T}\left( {{A\overset{\_}{x}} + {By}^{L} + b} \right)}} - {\eta^{T}{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)} + {\mu^{T}\left( {{G\left( \overset{\_}{y} \right)} - {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)}} \right)} + {\sum\limits_{i}\;{\ell_{i}(\pi)}}}}},\mspace{20mu}{where}$${\ell_{i}(\pi)} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu}\begin{bmatrix}{{B^{T}\pi} + {{\nabla_{y}^{T}{H\left( \overset{\_}{y} \right)}}\eta} +} \\{{\nabla_{y}^{T}{G\left( \overset{\_}{y} \right)}}\mu}\end{bmatrix}}_{i} \geq 0} \\{\begin{bmatrix}{{B^{T}\pi} + {{\nabla_{y}^{T}{H\left( \overset{\_}{y} \right)}}\eta} +} \\{{\nabla_{y}^{T}{G\left( \overset{\_}{y} \right)}}\mu}\end{bmatrix}_{i}\left\lbrack {y^{U} - y^{L}} \right\rbrack}_{i} & {{otherwise},}\end{matrix} \right.$and π, μ, η, and z, are new decision variables. Suppose that ( π, η, μ)is the optimal solution to the dual problem. By applying Lagrangianduality to the problem (6), the computing system 400 is configured toobtain

${{{\overset{\_}{\pi}}^{T}\left( {{A\;\overset{\_}{x}} + {By}^{L} + b} \right)} - {{\overset{\_}{\eta}}^{T}{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)} + {{\overset{\_}{\mu}}^{T}\left( {{G\left( \overset{\_}{y} \right)} - {\nabla_{y}{G\left( \overset{\_}{y} \right)}} - {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)}} \right)} + {\sum\limits_{i}{l_{i}\left( \overset{\_}{\pi} \right)}}} = {\sum\limits_{i}{\alpha_{i}^{*}.}}$Note that Σ_(i)l_(i)( π)≦0. Together with the strict positiveness ofΣ_(i)α_(i)*, it follows thatπ ^(T)(A x+By ^(L) +b)− η ^(T)∇_(y) H( y )( y−y ^(L))+ μ ^(T)(G( y)−∇_(y) G( y )( y−y ^(L)))>0.

Hence, the computing system 400 is configured to construct a first cutas follows:π ^(T)(A x+By ^(L) +b)− η ^(T)∇_(y) H( y )( y−y ^(L))+ μ ^(T)(G( y)−∇_(y) G( y )( y−y ^(L))≦0,  (7)which is an affine function of x. An affine function refers to a lineartransformation using a matrix and a vector.

The computing system 400 can be configured to create another Lagrangedual problem for the problem (6) as follows:

${{{\max\limits_{\pi,\mu,{\theta \geq 0},\eta}{\min\limits_{\alpha,{z \geq 0}}{\sum\limits_{i}\;\alpha_{i}}}} + {\pi^{T}\left( {{Bz} - \alpha + {A\overset{\_}{x}} + {By}^{L} + b} \right)} + {\eta^{T}\left( {{H\left( \overset{\_}{y} \right)} + {{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {z - \left( {\overset{\_}{y} - y^{L}} \right)} \right)}} \right)} + {\mu^{T}\left( {{G\left( \overset{\_}{y} \right)} + {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {z - \left( {\overset{\_}{y} - y^{L}} \right)} \right)}} \right)} + {\theta^{T}\left( {z - \left( {y^{U} - y^{L}} \right)} \right)}} = {{{\max\limits_{\pi,\mu,{\theta \geq 0},\eta}{\min\limits_{\alpha,{z \geq 0}}{\sum\limits_{i}\;{\alpha_{i}\left( {1 - \pi_{i}} \right)}}}} + {\left( {{B^{T}\pi} + {{\nabla_{y}^{T}{H\left( \overset{\_}{y} \right)}}\eta} + {{\nabla_{y}^{T}{G\left( \overset{\_}{y} \right)}}\mu} + \theta} \right)^{T}z} + {\pi^{T}\left( {{A\overset{\_}{x}} + {By}^{L} + b} \right)} - {\eta^{T}{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)} + {\mu^{T}\left( {{G\left( \overset{\_}{y} \right)} - {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)}} \right)} - {\theta^{T}\left( {y^{U} - y^{L}} \right)}} = {{\max\limits_{\underset{\underset{{{B^{T}\pi} + {{\nabla_{y}^{T}{G{(\overset{\_}{y})}}}\mu} + {{\nabla_{y}^{T}{H{(\overset{\_}{y})}}}\eta} + \theta} \geq 0}{\mu,{\theta \geq 0},\eta}}{1 \geq \pi_{i} \geq 0}}{\pi^{T}\left( {{A\overset{\_}{x}} + {By}^{L} + b} \right)}} - {\eta^{T}{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)} + {\mu^{T}\left( {{G\left( \overset{\_}{y} \right)} - {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)}} \right)} - {\theta^{T}\left( {y^{U} - y^{L}} \right)}}}},$where θ is a new decision variable.Again, suppose that ({circumflex over (π)}, {circumflex over (η)},{circumflex over (μ)}, {circumflex over (θ)}) is the optimal solution tothe dual problem. By solving the Lagrangian dual problem of the problem(6), the computing system is configured to obtain

${{{\hat{\pi}}^{T}\left( {{A\;\overset{\_}{x}} + {By}^{L} + b} \right)} - {{\hat{\eta}}^{T}{\nabla_{y}{H\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)} + {{\hat{\mu}}^{T}\left( {{G\left( \overset{\_}{y} \right)} - {{\nabla_{y}{G\left( \overset{\_}{y} \right)}}\left( {\overset{\_}{y} - y^{L}} \right)}} \right)} - {{\hat{\theta}}^{T}\left( {y^{U} - y^{L}} \right)}} = {{\sum\limits_{i}\alpha_{i}^{*}} > 0}$that implies a second cut as follows:{circumflex over (π)}^(T)(A x+By ^(L) +b)−{circumflex over (η)}^(T)∇_(y)H( y )( y−y ^(L))+{circumflex over (μ)}^(T)(G( y )−∇_(y) G( y )( y−y^(L)))−{circumflex over (θ)}^(T)(y ^(U) −y ^(L))≦0.  (8)

Motivated by equations (7) and (8), the computing system 400 isconfigured to obtain a third cut (e.g., equation (9)), provided that theoptimal result of (5) is strictly positive, i.e., (4) is infeasible.Denote ({tilde over (π)}, {tilde over (μ)}, {tilde over (η)}) by theLagrange multipliers in the nonlinear optimization problem (5). Thethird cut is as follows:{tilde over (π)}^(T)(Ax+By ^(L) +b)−{tilde over (η)}^(T)∇_(y) H( y )(y−y ^(L))+{tilde over (μ)}^(T)∇_(y) G( y )( y−y ^(L)))≦0.  (9)

By using these cuts (e.g., equations (7)-(9)), the computing system 400is configured to prune one or more results of the optimization problem(e.g., problem (5) or problem (6)). Returning to FIG. 1, at step 140,the computing system 400 is configured to solve the optimization problem(e.g., problem (5) or problem (6)) to obtain the pruned result. In oneembodiment, the master optimization problem with the pruned result is asfollows:

$\begin{matrix}\begin{matrix}\min & {f\left( u_{0} \right)} \\\; & {{g_{0}\left( {x_{0},u_{0}} \right)} \leq 0} \\\; & {{h_{0}\left( {x_{0},u_{0}} \right)} = 0} \\\; & {{{{Pu}_{0} + q} \leq 0},}\end{matrix} & (10)\end{matrix}$where Pμ₀+q≦0 includes the feasibility cuts (e.g., equations (7)-(9))with P is a matrix, q is a vector.

In other words, by running the method steps in FIG. 1, the computingsystem 400 is first configured to solve the problem (11) to get (x₀ ⁽¹⁾,u₀ ⁽¹⁾) (x₀ ⁽¹⁾ and u₀ ⁽¹⁾ are solutions obtained by solving theequation (11).)

$\begin{matrix}\begin{matrix}\min & {f\left( u_{0} \right)} \\\; & {{g_{0}\left( {x_{0},u_{0}} \right)} \leq 0} \\\; & {{h_{0}\left( {x_{0},u_{0}} \right)} = 0}\end{matrix} & (11)\end{matrix}$Then, for an iteration k=1, 2, . . . and each contingency c=1, . . . ,C, the computing system 400 is configured to solve the problem (5) for agiven (x₀ ^((k)), u₀ ^((k))). If Σ_(i)α_(i)>0, the computing system 400is configured to add the third cut (e.g., equation (9)) into the problem(10). If Σ_(i)α_(i)=0 for all c=1, . . . , C, the computing system isconfigured to terminate the method steps in FIG. 1. Otherwise, ifΣ_(i)α_(i)≠0, the computing system 400 is configured to solve theproblem (10) after adding the cut (e.g., equation (7) or (8) or (9)) toobtain (x₀ ^((k+1)), u₀ ^((k+1))). By solving problem (10), thecomputing system 400 is configured to output a result that specifies anamount of electric power to be generated to meet the one or morecustomer request under the contingency constraint and a lowest cost togenerate the amount of the electric power.

In one embodiment, the computing system is configured to utilize one ormore of these cuts (e.g., equations (7)-(9)) in an adaptive manner. Forexample, in one embodiment, if it is supposed that c(x)≦0 is a valid cutfor x, then c(x)−αc( x)≦0, for any α<1 is also a cut. By choosing aadaptively based on a constraint infeasibility measure for x, forexample, the measurement can be an optimal result of the problem (5),the quality of the solution of the optimization problem (1) can beimproved. The idea is that if the constraint infeasibility measure islarge (e.g. larger than 10, etc.), the computing system 400 isconfigured to select α close to 1, otherwise the computing system 400 isconfigured to select α close to 0.

FIG. 2 is a flow chart that describes method steps for solving theoptimization problem (1) that calculates amount of the electric power tobe generated and a total cost for generating the calculated amount ofthe electric power while meeting at least one contingency constraint andone or more customer request in one embodiment.

The computing system 400 is configured to generate a template to solvethe optimization problem (e.g., problem (1)). An example of the templateincludes, but is not limited to:

$\begin{matrix}{\min\limits_{x,y}{\left\{ {{{{F(x)} + {{G(y)}\text{:}\mspace{14mu} M\; x} + {Ny}} = d},{x \in X},{y \in Y}} \right\}.}} & (14)\end{matrix}$where xεP^(n), yεP^(m), MεP^(p×n), NεP^(p×m) and dεP^(p). Assume that Fand G are closed, convex, and differentiable functions, and X and Y areclosed and nonconvex sets.

The computing system 400 is configured to form an augmented Lagrangianfor the problem (14) as follows:

$\begin{matrix}{{{L_{\beta}\left( {x,y,\lambda} \right)} = {{F(x)} + {G(y)} + {\lambda^{T}\left( {{Mx} + {Ny} - d} \right)} + {\frac{1}{2}\beta{{{Mx} + {Ny} - d}}^{2}}}},\mspace{20mu}{{{where}\mspace{14mu}\beta} > 0.}} & (15)\end{matrix}$

A classical augmented Lagrangian multiplier method involves a jointoptimization and multiplier update step:(x ^(k+1) ,y ^(k+1)):=argmin_(xεX,yεY) L _(β)(x,y,λ ^(k))λ^(k+1)=λ^(k)+β(Mx ^(k+1) +Ny ^(k+1) −d).The computing system 400 is configured to iterate:x ^(k+1):=argmin_(xεX) L _(β)(x,y ^(k),λ^(k))y ^(k+1):=argmin_(yεY) L _(β)(x ^(k+1) ,y,λ ^(k))λ^(k+1)=λ^(k)+β(Mx ^(k+1) +Ny ^(k+1) −d),where x:=argmin_(xεΩ)f(x), means x* is the optimal solution of min f(x)s.t. xεΩ.

At step 210 in FIG. 2, in a further embodiment, the system is configuredto reformulate the optimization problem (1) based on the generatedtemplate. In particular, the optimization problem (1) can bereformulated, e.g., by introducing an auxiliary variable u_(0c) asfollows:

$\begin{matrix}{{\min\;{f\left( u_{0} \right)}}{{s.t.\mspace{14mu}{h_{0}\left( {x_{0},u_{0}} \right)}} = 0}{{g_{0}\left( {x_{0},u_{0}} \right)} \leq 0}{{h_{c}\left( {x_{c},u_{c}} \right)} = 0}{{g_{c}\left( {x_{c},u_{c}} \right)} \leq 0}{{{{u_{c}^{i} - u_{0\; c}^{i}}} \leq {\Delta\; u_{i}^{\max}}},{i \in G},{c = 1},\ldots\mspace{14mu},C}{{{u_{0}^{i} - u_{0\; c}^{i}} = 0},{i \in G},{c = 1},\ldots\mspace{14mu},{C.}}} & (16) \\{{{Consider}\mspace{14mu}{that}\text{:}}{{x:=\left( {x_{0},u_{0}} \right)},{y:=\left( {x_{1},u_{1},\ldots\mspace{14mu},x_{c},u_{c}} \right)}}{{{F(x)}:={f\left( u_{0} \right)}},{{G(y)}:=0}}{X:=\left\{ {{{\left( {x_{0},u_{0}} \right)\text{:}\mspace{14mu}{h_{0}\left( {x_{0},u_{0}} \right)}} = 0},{{g_{0}\left( {x_{0},u_{0}} \right)} \leq 0}} \right\}}{{Y:=\left( {Y_{1},\ldots\mspace{14mu},Y_{C}} \right)},{where}}\mspace{11mu}\;{{Y_{c}:=\begin{Bmatrix}{{{\left( {x_{c},u_{c},u_{0\; c}} \right)\text{:}\mspace{14mu}{h_{c}\left( {x_{c},u_{c}} \right)}} = 0},} \\{{{g_{c}\left( {x_{c},u_{c}} \right)} \leq 0},{{{u_{c}^{i} - u_{0\; c}^{i}}} \leq {\Delta\; u_{i}^{\max}}},{i \in G}}\end{Bmatrix}},{c = 1},\ldots\mspace{14mu},C}{{{{Mx} + {Ny}} = {{{d\text{:}\mspace{14mu} u_{0}} - u_{0\; c}} = 0}},{c = 1},\ldots\mspace{14mu},C}{{M:=\begin{bmatrix}1 & 0 & \ldots & 0 \\0 & 1 & \ldots & 0 \\\vdots & \; & \ddots & \vdots \\0 & \ldots & 0 & 1\end{bmatrix}},{N:={- \begin{bmatrix}1 & 0 & \ldots & 0 \\0 & 1 & \ldots & 0 \\\vdots & \; & \ddots & \vdots \\0 & \ldots & 0 & 1\end{bmatrix}}},{d = {\begin{bmatrix}0 \\\vdots \\0\end{bmatrix}.}}}} & (17)\end{matrix}$

Returning to FIG. 2, at step 220, the computing 400 is configured toform the Lagrangian for the reformulated optimization problem (16):

${L_{\beta}\left( {x,y,\lambda} \right)} = {{f\left( u_{0} \right)} + {\sum\limits_{c = 1}^{C}\;{\lambda_{c}\left( {u_{0} - u_{0\; c}} \right)}} + {\frac{\beta}{2}{\sum\limits_{c = 1}^{C}\;{{u_{0} - u_{0\; c}}}^{2}}}}$where  β > 0.The computing system 400 is configured to iterate, based on analternating direction method of multipliers, as follows:For k=1, 2, . . . .

$\left( {x_{0}^{({k + 1})},u_{0}^{({k + 1})}} \right) = {{{argmin}_{{({x_{0},u_{0}})} \in X}{f\left( u_{0} \right)}} + {\sum\limits_{c}\;{\lambda_{c}^{(k)}\left( {u_{0} - u_{0\; c}^{(k)}} \right)}} + {\frac{\beta}{2}{\sum\limits_{c}\;{{u_{0} - u_{0\; c}^{(k)}}}^{2}}}}$$\left( {x_{c}^{({k + 1})},u_{c}^{({k + 1})},u_{0\; c}^{({k + 1})}} \right) = {{{argmin}_{{({x_{c},{u_{c}u_{0\; c}}})} \in Y_{c}}{\sum\limits_{c}\;{\lambda_{c}^{(k)}\left( {u_{0}^{({k + 1})} - u_{0\; c}} \right)}}} + {\frac{\beta}{2}{\sum\limits_{c}\;{{u_{0}^{({k + 1})} - u_{0\; c}}}^{2}}}}$  c = 1, …  , C     λ_(c)^((k + 1)) = λ_(c)^((k)) + β(u₀^((k + 1)) − u_(0 c)^((k + 1))), c = 1, …  , C,The computing system solves the optimization problem (1) to determine(x₀, u₀), where x₀ are voltages, u₀ are the amount of an electric powergenerated at generators. The final iterate (x₀ ^((k+1)), u₀ ^((k+1)))provides (x₀, u₀).

Returning to FIG. 2, at step 230, the computing system 400 is configuredto initialize Lagrangian multipliers and decision variables according toeach contingency constraint: u_(0c) ⁽¹⁾, λ_(c) ⁽¹⁾, c=1, . . . , C.

At step 240, the computing system 400 is configured to solve thereformulated optimization problem (e.g., problem (16), “base caseoptimization problem” shown at step 240 in FIG. 2) with the initializedLagrangian multipliers and decision variables of contingencies:

$\left( {x_{0}^{({k + 1})},u_{0}^{({k + 1})}} \right) = {{{argmin}_{{({x_{0},u_{0}})} \in X}{f\left( u_{0} \right)}} + {\sum\limits_{c}\;{\lambda_{c}^{(k)}\left( {u_{0} - u_{0\; c}^{(k)}} \right)}} + {\frac{\beta}{2}{\sum\limits_{c}\;{{u_{0} - u_{0\; c}^{(k)}}}^{2}}}}$

At step 250, the computing system 400 is further configured to break thereformulated optimization problem (e.g., equation (16)) into one or moresub-problems, which can be run in parallel. The computing system 400 isconfigured to solve the one or more sub-problems with the initializedLagrangian multipliers and decision variables of the base case (e.g.,problem (16)), based on the formed augmented Lagrangian.

$\left( {x_{c}^{({k + 1})},u_{c}^{({k + 1})},u_{0\; c}^{({k + 1})}} \right) = {{{argmin}_{{({x_{c},{u_{c}u_{0\; c}}})} \in Y_{c}}{\sum\limits_{c}\;{\lambda_{c}^{(k)}\left( {u_{0}^{({k + 1})} - u_{0\; c}} \right)}}} + {\frac{\beta}{2}{\sum\limits_{c}\;{{u_{0}^{({k + 1})} - u_{0\; c}}}^{2}}}}$  c = 1, …  , C   

At step 260, the computing system 400 is configured to evaluate whethera solution of the one or more sub-problems meets a stopping criterion.The stopping criterion includes, but is not limited to: a gap between u₀^((k+1)) and u_(0c) ^((k+1)) is lower than a pre-determined thresholdfor all c=1, . . . , C.

At step 280, the computing system 400 is configured to output thesolution in response to determining that the solution meets the stoppingcriterion. The solution specifies ideal amount of electric power to begenerated to meet the customers' requests under the contingencyconstraint and a lowest cost to generate the ideal amount of theelectric power. At step 270, the computing system 400 is configured toupdate the Lagrangian multipliers based on the solutions from steps 270and 280, in response to determining that the solution does not meet thestopping criterion, and the process returns to step 240.

FIG. 3 is a flow chart that describes method steps for solving anoptimization problem that calculates amount of the electric power to begenerated and a total cost for generating the calculated amount of theelectric power while meeting at least one contingency constraint and oneor more customer request in one embodiment. At step 310, the computingsystem 400 is configured to build a tree data structure (not shown).Each node of the tree data structure describes a possible solution ofthe optimization problem (e.g., problem (1)). The system is configuredto compute, at each node of the built tree data structure, an upperbound and lower bound of a function (e.g., an objective function of theproblem (1)). The function determines an amount of electric power to begenerated to meet the one or more customer request under the contingencyconstraint and a cost for generating the amount of the electric power.The lower bound is calculated from Lagrange dual problem (i.e., Lagrangeduality). The upper bound is calculated from an interior-point method. Areference to Andreas Wachter, et al., “On the implementation of aninterior-point filter line-search algorithm for large-scale nonlinearprogramming,” Math. Program. Ser. A 106, pp. 25-57, Apr. 28, 2005,wholly incorporated by reference as set forth herein, describesinterior-point method in detail. At step 320, the computing system 400is configured to evaluate, at each node of the built tree datastructure, whether the computed upper bound and computed lower boundsatisfy a stopping criterion. The stopping criterion includes, but isnot limited to, one or more of: a gap between a smallest computed lowerbound and the computed upper bound is lower than a pre-determinedthreshold, and the tree data structure is empty. At step 370, thecomputing system 400 is configured to output the computed upper boundand the computed lower bound in response to determining that thecomputed upper bound and smallest computed lower bound satisfy thestopping criterion or the tree data structure is empty.

In one embodiment, the computing system 400 is configured to divide thefeasible set by using rectangular bisection or ellipsodial bisection toform the tree data structure. If an optimization problem min f(x) s.t.xεΩ, then SI is called the feasible set. Consider an electric powersystem with N nodes where P_(i) ^(G) and Q_(i) ^(G) are dispatchableactive and reactive powers, P_(i) ^(D) and Q_(i) ^(D) are active andreactive power demands at node i. Denote Γ by a set of electric powergenerators and Δ by a set of demand buses, whereas T is a set oftransmission lines. Every complex nodal voltage can be written inrectangular form V_(i)=e₁+if_(i), for all iεN, where i is the imaginaryunit, e_(i) is the real part of the complex nodal voltage V_(i) andf_(i) the imaginary part of the complex nodal voltage V_(i). In anelectric power system analysis, power system networks can be representedas graphs where electrical nodes correspond to graph nodes, transmissionlines are graph edges. In turn, each graph can be expressed as anadmittance matrix Y. The complex elements at row i and column j of thematrix Y are as follows: Y_(ij)=g_(ij)+ib_(ij), where G=(g_(ij)) is aconductance matrix, and B=(b_(ij)) is a susceptance matrix. Conductancematrix and susceptance matrix represent the physical characteristics,e.g. resistance and reactance, of an electric power network betweenelectric transmission lines i and j. From Kirchoffs law, electric powerflow equations at each node i can be formulated as the followingquadratic equations:

${{\sum\limits_{j \in N}\;\left( {{e_{i}\left( {{g_{ij}e_{j}} - {b_{ij}f_{j}}} \right)} + {f_{i}\left( {{g_{ij}f_{j}} + {b_{ij}e_{j}}} \right)}} \right)} = {P_{i}^{G} - P_{i}^{D}}},{{\sum\limits_{j \in N}\;\left( {{f_{i}\left( {{g_{ij}e_{j}} - {b_{ij}f_{j}}} \right)} - {e_{i}\left( {{g_{ij}f_{j}} + {b_{ij}e_{j}}} \right)}} \right)} = {Q_{i}^{G} - {Q_{i}^{D}.}}}$

An upper limit constraint on an active electric power flow in atransmission line (i, j) is represented as follows:P _(ij)=(e _(i) f _(j) −e _(j) f _(i))b _(i,j)+(e _(i) ² +f _(i) ² −e_(i) e _(j) −f _(i) f _(j))g _(ij) ≦P _(ij) ^(max),where P_(ij) ^(max) is the maximum active (MW) electic power flow in thetransmission line (i, j).

A node “1” is selected as a reference node, which implies from a zeroangular condition at this reference node that f₁=0. Taking into accountthe physical and operational limits to insure secure operating statesfor both the pre-contingency (normal case) constraints andpost-contingency constraints, this leads to the optimization problem:

$\begin{matrix}{{{\min\limits_{e_{i}^{c},f_{i}^{c},P_{i}^{G,c},Q_{i}^{G,c}}{\sum\limits_{i \in \Gamma}\;{c_{0\; i}\left( P_{i}^{G,0} \right)}^{2}}} + {c_{1\; i}P_{i}^{G,0}} + c_{2\; i}}{{subject}\mspace{14mu}{to}\mspace{14mu}\left( {{{{for}\mspace{14mu} c} = 0},\ldots\mspace{14mu},C} \right)\text{:}}} & (18) \\{{{\sum\limits_{j \in N}\;\left( {{e_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} + {f_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}} \right)} = {P_{i}^{G,c} - P_{i}^{D}}},{i \in N}} & (19) \\{{{\sum\limits_{j \in N}\;\left( {{f_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} - {e_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}} \right)} = {Q_{i}^{G,c} - Q_{i}^{D}}},{i \in N}} & (20) \\{{{{\left( {{e_{i}^{c}f_{j}^{c}} - {e_{j}^{c}f_{i}^{c}}} \right)b_{i,j}^{c}} + {\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2} - {e_{i}^{c}e_{j}^{c}} - {f_{i}^{c}f_{j}^{c}}} \right)g_{ij}^{c}}} \leq P_{ij}^{\max}},\mspace{85mu}{\left( {i,j} \right) \in T}} & (21) \\{f_{1}^{c} = 0} & (22) \\{{P_{i}^{\min} \leq P_{i}^{G,c} \leq P_{i}^{\max}},{i \in \Gamma}} & (23) \\{{Q_{i}^{\min} \leq Q_{i}^{G,c} \leq Q_{i}^{\max}},{i \in \Gamma}} & (24) \\{{\left( V_{i}^{\min} \right)^{2} \leq {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \leq \left( V_{i}^{\max} \right)^{2}},{i \in N}} & (25) \\{{{{P_{i}^{G,c} - P_{i}^{G,0}}} \leq {\Delta\; P_{i}}},{i \in \Gamma}} & (26) \\{{{{Q_{i}^{G,c} - Q_{i}^{G,0}}} \leq {\Delta\; Q_{i}}},{i \in \Gamma},} & (27)\end{matrix}$where P_(i) ^(min) and P_(i) ^(max) are active power generation limits,Q_(i) ^(min) and Q_(i) ^(max) are reactive power generation limits,V_(i) ^(min) and V_(i) ^(max) are voltage limits. ΔP_(i) and ΔQ_(i) arepre-determined maximal allowed variation of control variables, c_(0i),c_(1i), and c_(2i) are cost parameters, and index c represents acontingency c.

Let Ω be a feasible set defined by equations (19)-(27) and assume SI isnon-empty. An immediate result can be deduced is that SI is a compactand non-convex set. Note that the problem defined by (18)-(27) is acontinuous optimization problem with a convex quadratic objectivesubject to non-convex quadratic constraints. Decision variables isrequired to satisfy the non-convex quadratic constraints. Theoptimization problem (18)-(27) is equivalent to

${\min\mspace{14mu}{\sum\limits_{i \in \Gamma}\;{c_{0\; i}\left( P_{i}^{G} \right)}^{2}}} + {c_{1\; i}P_{i}^{G}} + c_{2\; i}$

subject to

the constraints (19)-(27)

$\begin{matrix}{{{\sum\limits_{i \in N}\;\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \right)} \leq {\sum\limits_{i \in N}\;\left( V_{i}^{\max} \right)^{2}}},\mspace{14mu}{c = 0},\ldots\mspace{14mu},{C.}} & (28)\end{matrix}$

The equations (23)-(24) represent constraints associated with therectangular branch and bound algorithm. The equation (28) represents aconstraint associated with the ellipsoidal branch and bound algorithm.The rectangular branch and bound algorithm and the ellipsoidal branchand bound algorithm are described in detail below.

Returning to FIG. 3, at step 330 the system is configured to apply atleast one branching procedure on the tree data structure in response todetermining that the computed upper bound and computed lower bound donot satisfy the stopping criteria. The branching procedure includes, butis not limited to: the rectangular bisection and the ellipsoidalbisection. The branching procedure in these branch and bound algorithmsis accomplished by dividing the feasible set to form the tree datastructure, e.g., by using either successive ellipsoidal bisections ofthe ellipsoidal constraint (e.g., equation (28)) or rectangularbisections of the rectangular constraints (e.g., equations (23)-(24)).

The following describes the ellipsoidal bisection in detail. Consider anellipsoid E with a center c in the form of:E={xεP ^(n):(x−c)^(T) B ⁻¹(x−c)≦1},  (29)where B is a symmetric, positive definite matrix. Given a nonzero vectorvεP^(n), the setsH ⁻ ={xεE:v ^(T) x≦v ^(T) c} and H ₊ ={xεE:v ^(T) x≧v ^(T) c}  (30)partition E into two sets of equal volume. Note that the hyperplane{x:v^(T)x=v^(T)c} passes through the center c of E. The centers c₊ andc⁻ and the matrix B_(±) of the ellipsoids E_(±) of minimum volumecontaining H_(±) are given as follows:

$\begin{matrix}{{c_{\pm} = {c \pm \frac{d}{n + 1}}},{B_{\pm} = {\frac{n^{2}}{n^{2} - 1}\left( {B - \frac{2\;{dd}^{T}}{n + 1}} \right)}},{d = \frac{Bv}{\sqrt{v^{T}{Bv}}}},} & (31)\end{matrix}$where n is the size of vector xεP^(n).A ratio R of the ellipsoid of E_(±) to the ellipsoid of E is

$\begin{matrix}{R = {\left( {\left( \frac{n}{n + 1} \right)^{n + 1}\left( \frac{n}{n - 1} \right)^{n - 1}} \right)^{\frac{1}{2}} < {\mathbb{e}}^{- \frac{1}{2n}} < 1.}} & (32)\end{matrix}$Thus R depends on the dimension n, but not on the vector v.

The following describes an ellipsoidal branch and bound algorithm thatuses an ellipsoidal bisection(s) in detail. For any ellipse E, definel(E) the lower bound of f(P^(Q,0)) over E∩Ω obtained from the optimalresult of the problem (44) described below. Assume that u_(k) is thebest upper bound value at k-th iteration. Likewise, l_(k) is thesmallest lower bound, and let S_(k) be the set containing direct-productellipsoids at the k-th iteration. Denote

${E^{c} = \left\{ {{\left( {e^{c},f^{c}} \right)\text{:}\mspace{14mu}{\sum\limits_{i \in N}\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \right)}} \leq {\sum\limits_{i \in N}\left( V_{i}^{\max} \right)^{2}}} \right\}},{c = 0},\ldots\mspace{14mu},C,{E_{0} = {E^{0} \times E^{1} \times \ldots \times {E^{C}.}}}$

[Ellipsoidal Branch and Bound Algorithm]

1. Set S₀={E₀}, evaluate l(E₀) and apply a local algorithm to get aninitial upper bound u₀, let l₀=l(E₀).

2. For k=0, 1, 2, . . . .

-   -   (a) If S_(k)=Ø or u_(k)=l_(k):=min{l(E):EεS_(k)}, then the        optimal solution of the problem (1) has been found    -   (b) Choose E_(k)εS_(k) such that l_(k)=l(E_(k)). Bisect the        ellipse associated with the largest diameter of E_(k). Use the        equations (29)-(31) to cover E_(k) with two new direct-product        ellipsoids denoted E_(k1) and E_(k2)    -   (c) Check if E_(ki)∩Ω=Ø set l(E_(ki))=∞;    -   otherwise evaluate l(E_(ki)) and compute a feasible point y_(ki)        of E_(ki)∩Ω, i=1, 2    -   (d) Let x_(k+1) denote a feasible point associated with the        lowest function value that has been generated up to this        iteration including the current step. We have        f(x_(k+1))≦f(y_(ki)) i=1, 2. If f(x_(k+1))<u_(k), then define        u_(k+1)=f(x_(k+1));    -   otherwise, u_(k+1)=u_(k)    -   (e) Set S_(k+1)={EεS_(k)∪{E_(k1)}∪{E_(k2)}:l(E)<u_(k+1),E≠E_(k)}

Since the union of two direct-product ellipsoids E_(k1)∪E_(k2) is alwaysgreater than E_(k), it is possible that at a given stage of theellipsoidal branch and bound algorithm the set E_(ki)∩Ω is empty. If so,the computing system is configured to prune E_(ki) from the tree datastructure as shown in step 2(c).

Suppose that in step 2(b) the vector v pointing along the axis is used.Then either the ellipsoidal branch and bound algorithm reaches anoptimal solution of the optimization problem (18)-(27) in a finitenumber of iterations, or every accumulation point of the sequence x_(k)is a solution of the problem (18)-(27).

The following describes rectangular bisection in detail. The branchingprocess in the this branch and bound algorithm is based on successiverectangular bisections of bound constraints for the active and reactivepowers: P^(G,c) and Q^(G,c)

$B^{c} = {\left\{ {\left( {P^{G,c},Q^{G,c}} \right)\text{:}\mspace{14mu}\begin{matrix}{{P_{i}^{\min} \leq P_{i}^{G,c} \leq P_{i}^{\max}},{i \in \Gamma}} \\{{Q_{i}^{\min} \leq Q_{i}^{G,c} \leq Q_{i}^{\max}},{i \in \Gamma}}\end{matrix}} \right\}.}$Denote B₀=B⁰×B¹× . . . ×B^(C). Consider a rectangle B in general formB={xεP^(n):a≦v≦b}, a point vεB, and an index jε{1, . . . , n}. Supposethat B is partitioned into two subrectangles B⁻ and B₊ determined by thehyperplane {x:x_(j)=v_(j)}:B ⁻ ={x:a _(j) ≦x _(j) v _(j) ,a _(i) ≦x _(i) ≦b _(i) for all i≠j},B ₊ ={x:v _(j) ≦x _(j) b _(j) ,a _(i) ≦x _(i) ≦b _(i) for all i≠j},A subdivision of B via (v, j) is a bisection of ratio α if an index jcorresponds to a longest side of B and v is a point of this side suchthatmin(v _(j) −a _(j) ,b _(j) −v _(j))=α(b _(j) −a _(j)).

Over a new generated rectangleB _(k) ={ P _(i) ^(min,c,k) ≦P _(i) ^(G,c) ≦ P _(i) ^(max,c,k) , Q _(i)^(min,c,k) ≦Q _(i) ^(G,c) ≦ Q _(i) ^(max,c,k), for all iεΓ,c=1, . . .,C},the Lagrange dual problem is given by

$\begin{matrix}{{{q\left( {\lambda,\beta,\mu,\gamma,\theta} \right)} = {\min\limits_{e^{c},f^{c},P^{G,c},Q^{G,c}}{L\left( {e^{c},f^{c},P^{G,c},Q^{G,c},\lambda,\beta,\mu,\gamma,\theta} \right)}}}\mspace{20mu}{{subject}\mspace{14mu}{to}}\mspace{20mu}{{f_{1}^{c} = 0},{c = 0},\ldots\mspace{14mu},C}\mspace{20mu}{{{\sum\limits_{i \in N}\left( V_{i}^{\min} \right)^{2}} \leq {\sum\limits_{i \in N}\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \right)} \leq {\sum\limits_{i \in N}\left( V_{i}^{\max} \right)^{2}}},{c = 0},\ldots\mspace{14mu},C}\mspace{20mu}{{P_{i}^{\min,c,k} \leq P_{i}^{G,c} \leq P_{i}^{\max,c,k}},{i \in \Gamma},{c = 0},\ldots\mspace{14mu},C}\mspace{20mu}{{Q_{i}^{\min,c,k} \leq Q_{i}^{G,c} \leq Q_{i}^{\max,c,k}},{i \in \Gamma},{c = 0},\ldots\mspace{14mu},C}\mspace{20mu}{{{{P_{i}^{G,c} - P_{i}^{G,0}}} \leq {\Delta\; P_{i}}},{i \in \Gamma},{c = 1},\ldots\mspace{14mu},C}\mspace{20mu}{{{{Q_{i}^{G,c} - Q_{i}^{G,0}}} \leq {\Delta\; Q_{i}}},{i \in \Gamma},{c = 1},\ldots\mspace{14mu},C,}} & (33)\end{matrix}$where L is defined in equation (37) below, the minimum can be calculatedby solving the problem (40) described below and the following

$\begin{matrix}{{{{minimize}_{e^{c},f^{c}}{\sum\limits_{i \in N}{\lambda_{i}^{c}\left( {\sum\limits_{j \in N}\left( {{e_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} + {f_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}} \right)} \right)}}} + {\sum\limits_{i \in N}{\beta_{i}^{c}\left( {\sum\limits_{j \in N}\left( {{f_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} - {e_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}} \right)} \right)}} + {\sum\limits_{{({i,j})} \in T}{\mu_{i,j}^{c}\left( {{\left( {{e_{i}^{c}f_{j}^{c}} - {e_{j}^{c}f_{i}^{c}}} \right)b_{ij}^{c}} + {\begin{pmatrix}{\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2} -} \\{{e_{i}^{c}e_{j}^{c}} - {f_{i}^{c}f_{j}^{c}}}\end{pmatrix}g_{ij}^{c}}} \right)}} + {\sum\limits_{i \in N}{\left( {\theta_{i}^{c} - \gamma_{i}^{c}} \right)\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \right)}}}\mspace{20mu}{{subject}\mspace{14mu}{to}}\mspace{20mu}{f_{i}^{c} = 0}\mspace{20mu}{{\sum\limits_{i \in N}\left( V_{i}^{\min} \right)^{2}} \leq {\sum\limits_{i \in N}\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \right)} \leq {\sum\limits_{i \in N}{\left( V_{i}^{\max} \right)^{2}.}}}} & (34)\end{matrix}$

Note that the definition of this q(λ, β, γ, θ) is slightly differentfrom the equation (38) described below, the computing system 400 isconfigured to add the lower bound constraints

$\begin{matrix}{{\sum\limits_{i \in N}\left( V_{i}^{\min} \right)^{2}} \leq {\sum\limits_{i \in N}{\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \right).}}} & (35)\end{matrix}$

Define the set F byF={(e ^(c) ,f ^(c) ,P ^(G,c) ,Q ^(G,c) ,c=0, . . .,C):(19),(20),(21),(22),(25),(26),(27) and (35)}.For any direct-product box B_(k), define l(B_(k)) the lower bound off(P^(Q)) over B_(k)∪F. Now, the rectangular branch and bound algorithmthat uses a rectangular bisection(s) can be stated as follows:

[Rectangular Branch and Bound Algorithm]

1. Set S₀={B₀}, evaluate l(B₀) and apply a local algorithm to get aninitial upper bound u₀, let l₀=l(B₀)

2. For k=0, 1, 2, . . . .

-   -   (a) If S_(k)=Ø or u_(k)=l_(k):=min{l(B):BεS_(k)}, then the        optimal solution of the problem (1) has been found    -   (b) Choose B_(k)εS_(k) such that l_(k)=l(B_(k)). Bisect B_(k)        into two direct-product boxes denoted B_(k1) and B_(k2)    -   (c) Check if B_(ki)∩F=Ø set l(B_(ki))=∞;    -   otherwise evaluate l(B_(ki)) and compute a feasible point y_(ki)        of B_(ki)∩F, i=1, 2    -   (d) Let x_(k+1) denote a feasible point associated with the        lowest function value that has been generated up to this        iteration including the current step. We have        f(x_(k+1))≦f(y_(ki)) i=1, 2. If f(x_(k+1))<u_(k), then define        u_(k+1)=f(x_(k+1));    -   otherwise, u_(k+1)=u_(k)    -   (e) Set        S_(k+1)={BεS_(k)∪{B_(k1)}∪{B_(k2)}:l(B)<u_(k+1),B≠B_(k)}.

At the root of the tree data structure, calculating the lower bounds forthe two branch and bound algorithms are identical. If an objectivefunction to be minimized is the active power losses of all branches

${{f\left( {e^{0},f^{0}} \right)} = {\sum\limits_{i \in N}{\sum\limits_{j \in N}{\left( g_{ij}^{0} \right)^{2}\left( {\left( {e_{i}^{0} - e_{j}^{0}} \right)^{2} + \left( {f_{i}^{0} - f_{j}^{0}} \right)^{2}} \right)}}}},$or any additional quadratic constraints are included, these two branchand bound algorithms still apply.

Returning to FIG. 3, at step 340, the computing system 400 is configuredto re-compute the lower bound in a region in the tree data structurebranched according to the applied branching procedure. At step 350, thecomputing system 400 is configured to update the upper bound in theregion in the tree data structure and the update of the upper bound isbased on the best feasible point (i.e., the point whose value is thesmallest among all generated feasible points) in the tree datastructure. At step 360, the computing system 400 is configured to deletenodes in the region based on the re-computed the lower bound and updatedupper bound. The system is configured to repeat steps 320-360 untilstopping criteria is met in which case method terminates at 370.

The following describes re-computing the lower bound for the ellipsoidalbranch and bound algorithm in detail, e.g., by using the Lagrange dualproblem. Assume that there exist an original feasible set Ω (i.e., theoriginal tree data structure where each node describes a possiblesolution of the problem (1)) and a direct-product (i.e., cartesianproduct) of ellipsoids E=E⁰×E¹× . . . ×E^(C). Consider that

$\begin{matrix}{{{minimize}\mspace{14mu}{f\left( P^{G,0} \right)}}{{{subject}\mspace{14mu}{to}\mspace{14mu}\left( {e^{c},f^{c},P^{G,c},{{Q^{G,c}\text{:}\mspace{14mu} c} = 0},\ldots\mspace{14mu},C} \right)} \in {\Omega\bigcap{E.}}}} & (36)\end{matrix}$

Then, define the Lagrangian L by incorporating all the constraints inthe problem (1) as follows:

$\begin{matrix}{{{L\left( {e^{c},f^{c},P^{G,c},Q^{G,c},\lambda,\beta,\mu,\gamma,\theta} \right)} = {{f\left( P^{G,0} \right)} + {\sum\limits_{{c = 0},\ldots\mspace{14mu},C}\begin{Bmatrix}{{\sum\limits_{i \in N}{\lambda_{i}^{c}\left( {{\sum\limits_{j \in N}\begin{pmatrix}{{e_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} +} \\{f_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}\end{pmatrix}} + P_{i}^{D}} \right)}} +} \\{{\sum\limits_{i \in N}{\beta_{i}^{c}\left( {{\sum\limits_{j \in N}\begin{pmatrix}{{f_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} -} \\{e_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}\end{pmatrix}} + Q_{i}^{D}} \right)}} +} \\{{\sum\limits_{{({i,j})} \in T}{\mu_{i,j}^{c}\begin{pmatrix}{{\left( {{e_{i}^{c}f_{j}^{c}} - {e_{j}^{c}f_{i}^{c}}} \right)b_{ij}^{c}} +} \\{{\begin{pmatrix}{\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2} -} \\{{e_{i}^{c}e_{j}^{c}} - {f_{i}^{c}f_{j}^{c}}}\end{pmatrix}g_{ij}^{c}} - P_{ij}^{\max}}\end{pmatrix}}} +} \\{{\sum\limits_{i \in N}{\gamma_{i}^{c}\left( {\left( V_{i}^{\min} \right)^{2} - \left( e_{i}^{c} \right)^{2} - \left( f_{i}^{c} \right)^{2}} \right)}} +} \\\begin{matrix}{{\sum\limits_{i \in N}{\theta_{i}^{c\;}\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2} - \left( V_{i}^{\max} \right)^{2}} \right)}} -} \\{{\sum\limits_{i \in \Gamma}{\lambda_{i}^{c}P_{i}^{G,c}}} - {\sum\limits_{i \in \Gamma}{\beta_{i}^{c}Q_{i}^{G,c}}}}\end{matrix}\end{Bmatrix}}}},\mspace{20mu}{{{where}\mspace{14mu}\lambda} = \left( {\lambda^{0},\lambda^{1},\ldots\mspace{14mu},\lambda^{C}} \right)},{\beta = {\left( {\beta^{0},\beta^{1},\ldots\mspace{14mu},\beta^{C}} \right)\mspace{14mu}{{etc}.}}}} & (37)\end{matrix}$

The Lagrange dual problem of the problem (1) is given by

$\begin{matrix}{{{q\left( {\lambda,\beta,\mu,\gamma,\theta} \right)} = {\min\limits_{e^{c},f^{c},P^{G,c},Q^{G,c}}{L\left( {e^{c},f^{c},P^{G,c},Q^{G,c},\lambda,\beta,\mu,\gamma,\theta} \right)}}}\mspace{20mu}{{subject}\mspace{14mu}{to}}\mspace{20mu}{{f_{1}^{c} = 0},{c = 0},\ldots\mspace{14mu},C}\mspace{20mu}{{\left( {e^{c},f^{c}} \right) \in E^{c}},{c = 0},\ldots\mspace{14mu},C}\mspace{20mu}{{P_{i}^{\min} \leq P_{i}^{G,c} \leq P_{i}^{\max}},{i \in \Gamma},{c = 0},\ldots\mspace{14mu},{C\mspace{20mu}{Q_{i}^{\min} \leq Q_{i}^{G,c} \leq Q_{i}^{\max}}},{i \in \Gamma},{c = 0},\ldots\mspace{14mu},{C\mspace{20mu}{{{P_{i}^{G,c} - P_{i}^{G,0}}} \leq {\Delta\; P_{i}}}},{i \in \Gamma},{c = 1},\ldots\mspace{14mu},{{C\mspace{20mu}{{Q_{i}^{G,c} - Q_{i}^{G,0}}}} \leq {\Delta\; Q_{i}}},{i \in \Gamma},{c = 1},\ldots\mspace{14mu},{C.}}} & (38)\end{matrix}$

For any value of the Lagrange multipliers {(λ, β, γ, θ):γ≦0, θ≧0}, thevalue of the dual function provides a lower bound on an optimal resultof the problem (1). To obtain the best lower bound for all possibleLagrange multipliers, the computing system is configured to solve thedual problem as follows:

$\begin{matrix}{{{maximize}_{\lambda,\beta,\mu,\gamma,\theta}\mspace{14mu}{q\left( {\lambda,\beta,\mu,\gamma,\theta} \right)}}{{{{subject}\mspace{14mu}{to}\mspace{14mu}\mu} \geq 0},{\gamma \geq 0},{\theta \geq 0.}}} & (39)\end{matrix}$

The computing system 400 is configured to decompose the problem (38)into C+2 sub-problems as follows:

$\begin{matrix}{{{{minimize}_{P^{G,c},Q^{G,c}}{f\left( P^{G,0} \right)}} - {\sum\limits_{{c = 0},\ldots\mspace{14mu},C}\left\{ {{\sum\limits_{i \in \Gamma}{\lambda_{i}^{c}P_{i}^{G,c}}} + {\sum\limits_{i \in \Gamma}{\beta_{i}^{c}Q_{i}^{G,c}}}} \right\}}}\mspace{20mu}{{subject}\mspace{14mu}{to}}\mspace{20mu}{{P_{i}^{\min} \leq P_{i}^{G,c} \leq P_{i}^{\max}},{i \in \Gamma},{c = 0},\ldots\mspace{14mu},{C\mspace{20mu}{Q_{i}^{\min} \leq Q_{i}^{G,c} \leq Q_{i}^{\max}}},{i \in \Gamma},{c = 0},\ldots\mspace{14mu},{C\mspace{20mu}{{{P_{i}^{G,c} - P_{i}^{G,0}}} \leq {\Delta\; P_{i}}}},{i \in \Gamma},{c = 1},\ldots\mspace{14mu},{{C\mspace{20mu}{{Q_{i}^{G,c} - Q_{i}^{G,0}}}} \leq {\Delta\; Q_{i}}},{i \in \Gamma},{c = 1},\ldots\mspace{14mu},C}\mspace{20mu}{and}} & (40) \\{{{{minimize}_{e^{c},f^{c}}{\sum\limits_{i \in N}{\lambda_{i}^{c}\left( {\sum\limits_{j \in N}\left( {{e_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} + {f_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}} \right)} \right)}}} + {\sum\limits_{i \in N}{\beta_{i}^{c}\left( {\sum\limits_{j \in N}\left( {{f_{i}^{c}\left( {{g_{ij}^{c}e_{j}^{c}} - {b_{ij}^{c}f_{j}^{c}}} \right)} - {e_{i}^{c}\left( {{g_{ij}^{c}f_{j}^{c}} + {b_{ij}^{c}e_{j}^{c}}} \right)}} \right)} \right)}} + {\sum\limits_{{({i,j})} \in T}{\mu_{i,j}^{c}\left( {{\left( {{e_{i}^{c}f_{j}^{c}} - {e_{j}^{c}f_{i}^{c}}} \right)b_{ij}^{c}} + {\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2} - {e_{i}^{c}e_{j}^{c}} - {f_{i}^{c}f_{j}^{c}}} \right)g_{ij}^{c}}} \right)}} + {\sum\limits_{i \in N}{\left( {\theta_{i}^{c} - \gamma_{i}^{c}} \right)\left( {\left( e_{i}^{c} \right)^{2} + \left( f_{i}^{c} \right)^{2}} \right)}}}\mspace{20mu}{{subject}\mspace{14mu}{to}}\mspace{20mu}{f_{1}^{c} = 0}\mspace{20mu}{\left( {e^{c},f^{c}} \right) \in {E^{c}.}}} & (41)\end{matrix}$

FIG. 4 illustrates an exemplary hardware configuration of the computingsystem 400 that runs the method steps described in FIGS. 1-3. Thehardware configuration preferably has at least one processor or centralprocessing unit (CPU) 411. The CPUs 411 are interconnected via a systembus 412 to a random access memory (RAM) 414, read-only memory (ROM) 416,input/output (I/O) adapter 418 (for connecting peripheral devices suchas disk units 421 and tape drives 440 to the bus 412), user interfaceadapter 422 (for connecting a keyboard 424, mouse 426, speaker 428,microphone 432, and/or other user interface device to the bus 412), acommunication adapter 434 for connecting the system 400 to a dataprocessing network, the Internet, an Intranet, a local area network(LAN), etc., and a display adapter 436 for connecting the bus 412 to adisplay device 438 and/or printer 439 (e.g., a digital printer of thelike).

As will be appreciated by one skilled in the art, aspects of the presentinvention may be embodied as a system, method or computer programproduct. Accordingly, aspects of the present invention may take the formof an entirely hardware embodiment, an entirely software embodiment(including firmware, resident software, micro-code, etc.) or anembodiment combining software and hardware aspects that may allgenerally be referred to herein as a “circuit,” “module” or “system.”Furthermore, aspects of the present invention may take the form of acomputer program product embodied in one or more computer readablemedium(s) having computer readable program code embodied thereon. In afurther embodiment, the computing system analyzes properties of theenterprise and market social networks to build features for predictivemodels of a propensity for a customer to buy a product, or to close adeal in a particular period of time.

Any combination of one or more computer readable medium(s) may beutilized. The computer readable medium may be a computer readable signalmedium or a computer readable storage medium. A computer readablestorage medium may be, for example, but not limited to, an electronic,magnetic, optical, electromagnetic, infrared, or semiconductor system,apparatus, or device, or any suitable combination of the foregoing. Morespecific examples (a non-exhaustive list) of the computer readablestorage medium would include the following: an electrical connectionhaving one or more wires, a portable computer diskette, a hard disk, arandom access memory (RAM), a read-only memory (ROM), an erasableprogrammable read-only memory (EPROM or Flash memory), an optical fiber,a portable compact disc read-only memory (CD-ROM), an optical storagedevice, a magnetic storage device, or any suitable combination of theforegoing. In the context of this document, a computer readable storagemedium may be any tangible medium that can contain, or store a programfor use by or in connection with a system, apparatus, or device runningan instruction.

A computer readable signal medium may include a propagated data signalwith computer readable program code embodied therein, for example, inbaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including, but not limited to,electro-magnetic, optical, or any suitable combination thereof. Acomputer readable signal medium may be any computer readable medium thatis not a computer readable storage medium and that can communicate,propagate, or transport a program for use by or in connection with asystem, apparatus, or device running an instruction.

Program code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wireline, optical fiber cable, RF, etc., or any suitable combination ofthe foregoing.

Computer program code for carrying out operations for aspects of thepresent invention may be written in any combination of one or moreprogramming languages, including an object oriented programming languagesuch as Java, Smalltalk, C++ or the like and conventional proceduralprogramming languages, such as the “C” programming language or similarprogramming languages. The program code may run entirely on the user'scomputer, partly on the user's computer, as a stand-alone softwarepackage, partly on the user's computer and partly on a remote computeror entirely on the remote computer or server. In the latter scenario,the remote computer may be connected to the user's computer through anytype of network, including a local area network (LAN) or a wide areanetwork (WAN), or the connection may be made to an external computer(for example, through the Internet using an Internet Service Provider).

Aspects of the present invention are described below with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems) and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer program instructions. These computer program instructions maybe provided to a processor of a general purpose computer, specialpurpose computer, or other programmable data processing apparatus toproduce a machine, such that the instructions, which run via theprocessor of the computer or other programmable data processingapparatus, create means for implementing the functions/acts specified inthe flowchart and/or block diagram block or blocks. These computerprogram instructions may also be stored in a computer readable mediumthat can direct a computer, other programmable data processingapparatus, or other devices to function in a particular manner, suchthat the instructions stored in the computer readable medium produce anarticle of manufacture including instructions which implement thefunction/act specified in the flowchart and/or block diagram block orblocks.

The computer program instructions may also be loaded onto a computer,other programmable data processing apparatus, or other devices to causea series of operational steps to be performed on the computer, otherprogrammable apparatus or other devices to produce a computerimplemented process such that the instructions which run on the computeror other programmable apparatus provide processes for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof code, which comprises one or more operable instructions forimplementing the specified logical function(s). It should also be notedthat, in some alternative implementations, the functions noted in theblock may occur out of the order noted in the figures. For example, twoblocks shown in succession may, in fact, be run substantiallyconcurrently, or the blocks may sometimes be run in the reverse order,depending upon the functionality involved. It will also be noted thateach block of the block diagrams and/or flowchart illustration, andcombinations of blocks in the block diagrams and/or flowchartillustration, can be implemented by special purpose hardware-basedsystems that perform the specified functions or acts, or combinations ofspecial purpose hardware and computer instructions.

What is claimed is:
 1. A method for determining an amount of an electricpower to generate in an electric power system and determining a totalcost for generating the amount of electric power while satisfying atleast one contingency constraint and one or more customer request, themethod comprising: creating, by using a computing system including atleast one memory device and at least one processor connected to thememory device, an optimization problem for calculating the amount of theelectric power to be generated and a total cost for generating thecalculated amount of the electric power while meeting the at least onecontingency constraint and the one or more customer request, the atleast one contingency constraint including one or more of: atransmission line failure, a power generator failure, a transformerfailure, a structure of a power distribution network, and a topology ofthe power distribution network, wherein said optimization problem isbased on an electric optimal power flow problem according to:min f(u ₀)s.t.h _(c)(x _(c) ,u _(c))=0g _(c)(x _(c) ,u _(c))≦0|u _(c) ^(i) −u ₀ ^(i) |≦Δu _(i) ^(max) ,iεG,c=0, . . . ,C, where c=0represents an electric power system having no occurrence of contingencyconstraint, c=1, . . . , C, is a variable representing a contingencyconstraint, x_(c) is a vector of state variables for a c-thconfiguration, u_(c) is a vector of control variables, andf(u₀)=Σ_(iεG)(c_(0i)(u₀ ^(i))²+c_(1i)u₀ ^(i)+c_(2i)) is a generationcost with cost parameters c_(0i), c_(1i), c_(2i)≧0, where G is a set ofgenerators, and C is the number of contingency constraints, where Δu_(i)^(max) is a pre-determined maximal allowed variation of controlvariables, and h_(c) and g_(c) are respective operational constraintsincluding AC power flow balance equations; running, by using thecomputing system, the optimization problem in real-time; and outputting,from the optimization problem run by the computing system, an outputspecifying the calculated amount of the electric power and the totalcost to generate the calculated amount of the electric power.
 2. Themethod according to claim 1, wherein the step of running theoptimization problem includes: solving, by the computing system, theoptimization problem without considering the at least one contingencyconstraint; checking, by the computing system, whether a result of thecomputed optimization problem satisfies the at least one contingencyconstraint; outputting, by the computing system, the result in responseto determining that the result satisfies the at least one contingencyconstraint; pruning, by computing system, in response to determiningthat the result does not satisfy the at least one contingencyconstraint, one or more results of the optimization problem that do notsatisfy the at least one contingency constraint; and solving, by thecomputing system, the optimization problem with the pruned results. 3.The method according to claim 1, wherein the step of running theoptimization problem includes: generating, by the computing system, atemplate to solve the optimization problem; reformulating, by thecomputing system, the optimization problem based on the generatedtemplate; forming, by the computing system, an augmented Lagrangian forthe reformulated optimization problem; initializing, by the computingsystem, Lagrangian multipliers and decision variables according to theat least one contingency constraint; and solving, by the computingsystem, the reformulated optimization problem with the initializedLagrangian multipliers and decision variables, based on an alternatingdirection method of multipliers.
 4. The method according to claim 3,further comprising: breaking the reformulated optimization problem intoone or more sub-problems; solving the one or more sub-problems with theinitialized Lagrangian multipliers and decision variables, based on theformed augmented Lagrangian; evaluating whether a solution of the one ormore sub-problems meets a stopping criterion; outputting the solution inresponse to determining that the solution meets the stopping criterion;and updating the Lagrangian multipliers based on the solution, inresponse to determining that the solution does not meet the stoppingcriterion.
 5. The method according to claim 1, wherein the step ofrunning the optimization problem includes: building a tree datastructure, each node of the tree data structure describing a possiblesolution of the optimization problem; computing, at each node of thebuilt tree data structure, an upper bound and lower bound of anobjective function for determining the cost for generating the electricpower, the lower bound being calculated from Lagrange duality, the upperbound being calculated from an interior-point method; evaluating, ateach node of the built tree data structure, whether the computed upperbound and computed lower bound satisfy a stopping criterion; outputtingthe computed upper bound and the computed lower bound in response todetermining that the computed upper bound and computed lower boundsatisfy the stopping criterion; applying at least one branchingprocedure on the tree data structure in response to determining that thecomputed upper bound and computed lower bound do not satisfy thestopping criteria; re-computing the lower bound in a region in the treedata structure branched according to the applied branching procedure;updating the upper bound in the region in the tree data structure;deleting nodes in the region based on the re-computed the lower boundand updated upper bound or an intersection of the branched region and anoriginal feasible set, the original feasible set referring to possiblesolutions of the optimization problem satisfying the at least onecontingency constraint; and repeating the step of evaluating, the stepof the outputting, the step of applying, the step of re-computing, thestep of updating, and the step of deleting.
 6. The method according toclaim 5, wherein the stopping criterion includes one or more of: a gapbetween a smallest computed lower bound and the computed upper bound islower than a pre-determined threshold, or the tree data structure isempty.
 7. The method according to claim 5, wherein the at least onebranching procedure includes one or more of: ellipsoidal bisection andrectangular bisection.
 8. The method according to claim 1, wherein theone or more customer request includes: a demand or load for a certainamount of the electric power.
 9. A system for determining an amount ofan electric power to generate in an electric power system anddetermining a total cost for generating the amount of electric powerwhile satisfying at least one contingency constraint and one or morecustomer request, the system comprising: at least one memory device; atleast one processor connected to the memory device, wherein theprocessor is configured to: create an optimization problem forcalculating an amount of the electric power to be generated and a totalcost for generating the calculated amount of the electric power whilemeeting the at least one contingency constraint and the one or morecustomer request, the at least one contingency constraint including oneor more of: a transmission line failure, a power generator failure, atransformer failure, a structure of a power distribution network, and atopology of the power distribution network, wherein said optimizationproblem is based on an electric optimal power flow problem according to:min f(u ₀)s.t.h _(c)(x _(c) ,u _(c))=0g _(c)(x _(c) ,u _(c))≦0|u _(c) ^(i) −u ₀ ^(i) |≦Δu _(i) ^(max) ,iεG,c=0, . . . ,C, where c=0represents an electric power system having no occurrence of contingencyconstraint, c=1, . . . , C, is a variable representing a contingencyconstraint, x_(c) is a vector of state variables for a c-thconfiguration, u_(c) is a vector of control variables, andf(u₀)=Σ_(iεG)(c_(0i)(u₀ ^(i))²+c_(1i)u₀ ^(i)+c_(2i)) is a generationcost with cost parameters c_(0i), c_(1i), c_(2i)≧0, where G is a set ofgenerators, and C is the number of contingency constraints, where Δu_(i)^(max) is a pre-determined maximal allowed variation of controlvariables, and h_(c) and g_(c) are respective operational constraintsincluding AC power flow balance equations; run the optimization problemin real-time; and output, from the optimization problem, an outputspecifying the calculated amount of the electric power and the totalcost to generate the calculated amount of the electric power.
 10. Thesystem according to claim 9, wherein to run the optimization problem,the processor is further configured to: solve the optimization problemwithout considering the at least one contingency constraint; checkwhether a result of the computed optimization problem satisfies the atleast one contingency constraint; output the result in response todetermining that the result satisfies the at least one contingencyconstraint; prune, in response to determining that the result does notsatisfy the at least one contingency constraint, one or more results ofthe optimization problem that do not satisfy the at least onecontingency constraint; and solve the optimization problem with thepruned results.
 11. The system according to claim 9, wherein to run theoptimization problem, the processor is further configured to: generate atemplate to solve the optimization problem; reformulate the optimizationproblem based on the generated template; form an augmented Lagrangianfor the reformulated optimization problem; initialize Lagrangianmultipliers and decision variables according to the at least onecontingency constraint; and solve the reformulated optimization problemwith the initialized Lagrangian multipliers and decision variables,based on an alternating direction method of multipliers.
 12. The systemaccording to claim 11, wherein the processor is further configured to:break the reformulated optimization problem into one or moresub-problems; solve the one or more sub-problems with the initializedLagrangian multipliers and decision variables, based on the formedaugmented Lagrangian; evaluate whether a solution of the one or moresub-problems meets a stopping criterion; outputting the solution inresponse to determining that the solution meets the stopping criterion;and update the Lagrangian multipliers based on the solution, in responseto determining that the solution does not meet the stopping criterion.13. The system according to claim 9, wherein to run the optimizationproblem, the processor is further configured to: build a tree datastructure, each node of the tree data structure describing a possiblesolution of the optimization problem; compute, at each node of the builttree data structure, an upper bound and lower bound of an objectivefunction, the objective function determining the cost for generating theelectric power, the lower bound being calculated from Lagrange duality,the upper bound being calculated from an interior-point method;evaluate, at each node of the built tree data structure, whether thecomputed upper bound and computed lower bound satisfy a stoppingcriterion; output the computed upper bound and the computed lower boundin response to determining that the computed upper bound and computedlower bound satisfy the stopping criterion; apply at least one branchingprocedure on the tree data structure in response to determining that thecomputed upper bound and computed lower bound do not satisfy thestopping criteria; re-compute the lower bound in a region in the treedata structure branched according to the applied branching procedure;update the upper bound in the region in the tree data structure; deletenodes in the region based on the re-computed the lower bound and updatedupper bound or an intersection of the branched region and an originalfeasible set, the original feasible set referring to possible solutionsof the optimization problem satisfying the at least one contingencyconstraint; and repeat the evaluate, the output, the apply, there-compute, the update, and the delete.
 14. The system according toclaim 13, wherein the stopping criterion includes one or more of: a gapbetween a smallest computed lower bound and the computed upper bound islower than a pre-determined threshold, or the tree data structure isempty.
 15. The system according to claim 13, wherein the at least onebranching procedure includes one or more of: ellipsoidal bisection andrectangular bisection.
 16. The system according to claim 9, wherein theone or more customer request includes: a demand or load for a certainamount of the electric power.
 17. A computer program product fordetermining an electricity dispatch plan of an electric power systemwith a total cost for determining an amount of an electric power togenerate in an electric power system and determining a total cost forgenerating the amount of electric power while satisfying at least onecontingency constraint and one or more customer request, the computerprogram product comprising a non-transitory storage medium readable by aprocessing circuit and storing instructions run by the processingcircuit for performing a method, the method comprising: creating anoptimization problem for calculating an amount of the electric power tobe generated and a total cost for generating the calculated amount ofthe electric power while meeting the at least one contingency constraintand the one or more customer request, the at least one contingencyconstraint including one or more of: a transmission line failure, apower generator failure, a transformer failure, a structure of a powerdistribution network, and a topology of the power distribution network,wherein said optimization problem is based on an electric optimal powerflow problem according to:min f(u ₀)s.t.h _(c)(x _(c) ,u _(c))=0g _(c)(x _(c) ,u _(c))≦0|u _(c) ^(i) −u ₀ ^(i) |≦Δu _(i) ^(max) ,iεG,c=0, . . . ,C, where c=0represents an electric power system having no occurrence of contingencyconstraint, c=1, . . . , C, is a variable representing a contingencyconstraint, x_(c) is a vector of state variables for a c-thconfiguration, u_(c) is a vector of control variables, andf(u₀)=Σ_(iεG)(c_(0i)(u₀ ^(i))²+c_(1i)u₀ ^(i)+c_(2i)) is a generationcost with cost parameters c_(0i), c_(1i), c_(2i)≧0, where G is a set ofgenerators, and C is the number of contingency constraints, where Δu_(i)^(max) is a pre-determined maximal allowed variation of controlvariables, and h_(c) and g_(c) are respective operational constraintsincluding AC power flow balance equations; running the optimizationproblem in real-time; and outputting an output specifying the calculatedamount of the electric power and the total cost to generate thecalculated amount of the electric power.
 18. The computer programproduct according to claim 17, wherein the step of running theoptimization problem includes: solving the optimization problem withoutconsidering the at least one contingency constraint; check whether aresult of the computed optimization problem satisfies the at least onecontingency constraint; output the result in response to determiningthat the result satisfies the at least one contingency constraint;prune, in response to determining that the result does not satisfy theat least one contingency constraint, one or more results of theoptimization problem that do not satisfy the at least one contingencyconstraint; and solve the optimization problem with the pruned results.19. The computer program product according to claim 17, wherein the stepof running the optimization problem includes: generating a template tosolve the optimization problem; reformulating the optimization problembased on the generated template; forming an augmented Lagrangian for thereformulated optimization problem; initializing Lagrangian multipliersand decision variables according to the at least one contingencyconstraint; and solving the reformulated optimization problem with theinitialized Lagrangian multipliers and decision variables, based on analternating direction method of multipliers.
 20. The computer programproduct according to claim 19, wherein the method further comprises:breaking the reformulated optimization problem into one or moresub-problems; solving the one or more sub-problems with the initializedLagrangian multipliers and decision variables, based on the formedaugmented Lagrangian; evaluating whether a solution of the one or moresub-problems meets a stopping criterion; outputting the solution inresponse to determining that the solution meets the stopping criterion;and updating the Lagrangian multipliers based on the solution, inresponse to determining that the solution does not meet the stoppingcriterion.
 21. The computer program product according to claim 17,wherein the step of running the optimization problem includes: buildinga tree data structure, each node of the tree data structure describing apossible solution of the optimization problem; computing, at each nodeof the built tree data structure, an upper bound and lower bound of anobjective function, the objective function determining the cost forgenerating the electric power, the lower bound being calculated fromLagrange duality, the upper bound being calculated from aninterior-point method; evaluating, at each node of the built tree datastructure, whether the computed upper bound and computed lower boundsatisfy a stopping criterion; outputting the computed upper bound andthe computed lower bound in response to determining that the computedupper bound and computed lower bound satisfy the stopping criterion;applying at least one branching procedure on the tree data structure inresponse to determining that the computed upper bound and computed lowerbound do not satisfy the stopping criteria; re-computing the lower boundin a region in the tree data structure branched according to the appliedbranching procedure; updating the upper bound in the region in the treedata structure; deleting nodes in the region based on the re-computedthe lower bound and updated upper bound or an intersection of thebranched region and an original feasible set, the original feasible setreferring to possible solutions of the optimization problem satisfyingthe at least one contingency constraint; and repeating the step ofevaluating, the step of the outputting, the step of applying, the stepof re-computing, the step of updating, and the step of deleting.
 22. Thecomputer program product according to claim 21, wherein the stoppingcriterion includes one or more of: a gap between a smallest computedlower bound and the computed upper bound is lower than a pre-determinedthreshold, or the tree data structure is empty.
 23. The computer programproduct according to claim 21, wherein the at least one branchingprocedure includes one or more of: ellipsoidal bisection and rectangularbisection.
 24. The computer program product according to claim 17,wherein the one or more customer request includes: a demand or load fora certain amount of the electric power.